HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdslj1i Unicode version

Theorem mdslj1i 22845
Description: Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdslle1.1  |-  A  e. 
CH
mdslle1.2  |-  B  e. 
CH
mdslle1.3  |-  C  e. 
CH
mdslle1.4  |-  D  e. 
CH
Assertion
Ref Expression
mdslj1i  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )

Proof of Theorem mdslj1i
StepHypRef Expression
1 ssin 3352 . . . . 5  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
21bicomi 195 . . . 4  |-  ( A 
C_  ( C  i^i  D )  <->  ( A  C_  C  /\  A  C_  D
) )
3 mdslle1.3 . . . . . 6  |-  C  e. 
CH
4 mdslle1.4 . . . . . 6  |-  D  e. 
CH
5 mdslle1.1 . . . . . . 7  |-  A  e. 
CH
6 mdslle1.2 . . . . . . 7  |-  B  e. 
CH
75, 6chjcli 21982 . . . . . 6  |-  ( A  vH  B )  e. 
CH
83, 4, 7chlubi 21996 . . . . 5  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  <->  ( C  vH  D )  C_  ( A  vH  B ) )
98bicomi 195 . . . 4  |-  ( ( C  vH  D ) 
C_  ( A  vH  B )  <->  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )
102, 9anbi12i 681 . . 3  |-  ( ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) )  <->  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )
11 simpr 449 . . . . . . . . . 10  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  B  MH*  A )
12 simpl 445 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  C )
13 simpl 445 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  C  C_  ( A  vH  B
) )
145, 6, 33pm3.2i 1135 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  C  e. 
CH )
15 dmdsl3 22841 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
1614, 15mpan 654 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  A )  =  C )
1711, 12, 13, 16syl3an 1229 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
183, 6chincli 21985 . . . . . . . . . . 11  |-  ( C  i^i  B )  e. 
CH
194, 6chincli 21985 . . . . . . . . . . 11  |-  ( D  i^i  B )  e. 
CH
2018, 19chub1i 21994 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
2118, 19chjcli 21982 . . . . . . . . . . 11  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH
2218, 21, 5chlej1i 21998 . . . . . . . . . 10  |-  ( ( C  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  ->  ( ( C  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
2320, 22mp1i 13 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
2417, 23eqsstr3d 3174 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  C  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
25 simpr 449 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  D )
26 simpr 449 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  D  C_  ( A  vH  B
) )
275, 6, 43pm3.2i 1135 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
28 dmdsl3 22841 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  D  e.  CH )  /\  ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
2927, 28mpan 654 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B
) )  ->  (
( D  i^i  B
)  vH  A )  =  D )
3011, 25, 26, 29syl3an 1229 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
3119, 18chub2i 21995 . . . . . . . . . 10  |-  ( D  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
3219, 21, 5chlej1i 21998 . . . . . . . . . 10  |-  ( ( D  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  ->  ( ( D  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3331, 32mp1i 13 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3430, 33eqsstr3d 3174 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  D  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3524, 34jca 520 . . . . . . 7  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  /\  D  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) ) )
3621, 5chjcli 21982 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  e.  CH
373, 4, 36chlubi 21996 . . . . . . 7  |-  ( ( C  C_  ( (
( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  /\  D  C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )  <->  ( C  vH  D )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3835, 37sylib 190 . . . . . 6  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  vH  D )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
39 ssrin 3355 . . . . . 6  |-  ( ( C  vH  D ) 
C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A
)  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B ) )
4038, 39syl 17 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B ) )
41 simpl 445 . . . . . 6  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  A  MH  B )
42 ssrin 3355 . . . . . . . 8  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( C  i^i  B ) )
4342, 20syl6ss 3152 . . . . . . 7  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
4443adantr 453 . . . . . 6  |-  ( ( A  C_  C  /\  A  C_  D )  -> 
( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
45 inss2 3351 . . . . . . . 8  |-  ( C  i^i  B )  C_  B
46 inss2 3351 . . . . . . . 8  |-  ( D  i^i  B )  C_  B
4718, 19, 6chlubi 21996 . . . . . . . . 9  |-  ( ( ( C  i^i  B
)  C_  B  /\  ( D  i^i  B ) 
C_  B )  <->  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B )
4847bicomi 195 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  C_  B 
<->  ( ( C  i^i  B )  C_  B  /\  ( D  i^i  B ) 
C_  B ) )
4945, 46, 48mpbir2an 891 . . . . . . 7  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
5049a1i 12 . . . . . 6  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  ( D  i^i  B ) )  C_  B )
515, 6, 213pm3.2i 1135 . . . . . . 7  |-  ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )
52 mdsl3 22842 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
) )  ->  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B )  =  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
5351, 52mpan 654 . . . . . 6  |-  ( ( A  MH  B  /\  ( A  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
)  ->  ( (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5441, 44, 50, 53syl3an 1229 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5540, 54sseqtrd 3175 . . . 4  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
56553expb 1157 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5710, 56sylan2b 463 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
583, 4, 6lediri 22062 . . 3  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  (
( C  vH  D
)  i^i  B )
5958a1i 12 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  ( ( C  vH  D )  i^i 
B ) )
6057, 59eqssd 3157 1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    i^i cin 3112    C_ wss 3113   class class class wbr 3983  (class class class)co 5778   CHcch 21455    vH chj 21459    MH cmd 21492    MH* cdmd 21493
This theorem is referenced by:  mdslmd1lem1  22851  mdslmd1lem2  22852
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cc 8015  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771  ax-hilex 21525  ax-hfvadd 21526  ax-hvcom 21527  ax-hvass 21528  ax-hv0cl 21529  ax-hvaddid 21530  ax-hfvmul 21531  ax-hvmulid 21532  ax-hvmulass 21533  ax-hvdistr1 21534  ax-hvdistr2 21535  ax-hvmul0 21536  ax-hfi 21604  ax-his1 21607  ax-his2 21608  ax-his3 21609  ax-his4 21610  ax-hcompl 21727
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-omul 6438  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-acn 7529  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-seq 10999  df-exp 11057  df-hash 11290  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-clim 11913  df-rlim 11914  df-sum 12110  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-cn 16905  df-cnp 16906  df-lm 16907  df-haus 16991  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cfil 18629  df-cau 18630  df-cmet 18631  df-grpo 20804  df-gid 20805  df-ginv 20806  df-gdiv 20807  df-ablo 20895  df-subgo 20915  df-vc 21048  df-nv 21094  df-va 21097  df-ba 21098  df-sm 21099  df-0v 21100  df-vs 21101  df-nmcv 21102  df-ims 21103  df-dip 21220  df-ssp 21244  df-ph 21337  df-cbn 21388  df-hnorm 21494  df-hba 21495  df-hvsub 21497  df-hlim 21498  df-hcau 21499  df-sh 21732  df-ch 21747  df-oc 21777  df-ch0 21778  df-shs 21833  df-chj 21835  df-md 22806  df-dmd 22807
  Copyright terms: Public domain W3C validator